Following [1]}, we equip \(\mathbb {R}^{\mathbb {N}}\)  with the product topology, and \(\pi _n:=(p_1,\dots ,p_n)\)  the canonical projections. The space \(AC_w([0,T];\mathbb {R}^\mathbb {N})\)  consists of curves \(\eta \)  such that \(p_i\circ \eta \in AC[0,T]\)  for all \(i\in \mathbb {N}\) . Note that both \(\mathbb {R}^\mathbb {N}\)  and \(C([0,T];\mathbb {R}^\mathbb {N})\)  are Polish spaces. Moreover, let \(|\cdot |_{\infty }\)  be the uniform norm on \(\mathbb {R}^{\mathbb {N}}\) .
